What is the radical ideal of $(u^2v-a^3,uv^2-b^3,uv-ab)$ in $\mathbb{C}[u,v,a,b]?$ Above all, to learn how to fish, what would be code that I can use to get the radical? I have not worked with Macaulay2 (computational algebra software) before, so what is a good reference to learn about?

All: I had studied abstract algebra long time ago. Now, I would like to review some material, particularly about Galois theory (and its application). Can anyone recommend an abstract algebra book which cover Galois theory (and its applications)? I have been a software engineer for past many years. Ideally, I would like an algebra book […]

This is a question in the mathematical software called GAP: What is the command for displaying all the generators of a given group? I have been searching around but yet not found anything helpful, so I am hoping I will get a quick response here.

Does it exist any program (for linux) which can generate a nice Cayley graph of any $\mathbb Z_n$? (If it’s possible to create such a graph at all, that is.) (where perhaps $n ≤ 100$ or something like that)

Question: How do I substitute a value into a polynomial in GAP? So, if I start off with the following: x:=Indeterminate(Integers,”x”); f:=x^2+3; I have $f$ as the polynomial $x^2+3$ over the integers. How can I find, say, $f(100)$? There should be a simple one line answer, but I can’t seem to find it in the […]

I’m looking for a FLOSS application (Windows or Ubuntu but preferably both) that can help me visualize matrix transformations in 2- and 3-space. So I’d like to be able to enter a vector or matrix, see it in 2-space or 3-space, enter a transformation vector or matrix, and see the result. For example, enter a […]

if $h$ divides $\# G$, then it is not necessarily that $G$ has a subgroup of size $h$ which can be related to Lagrange Theorem $[G:H]=\frac{\#G}{\# H}$ in the case when the subgroup exists, however it is not generally true such as in the case of $\mathbb A_4$ with order of 12 and having no […]

Suppose $$f_1=-4x^4y^2z^2+y^6+3z^5,$$ $$f_2=-4x^2y^2z^2+y^6+3z^5,$$ $$f_3=4x^4y^2z^2+y^6+3z^5,$$ $$f_4=4x^2y^2z^2+y^6+3z^5$$ and $$I=\langle xz-y^2,x^3-z^2\rangle\subset\mathbb C[x,y,z].$$ Is $f_i\in I?$ The answer is Yes in some cases. The question can be checked with Macaulay 2: when the remainder is zero with respect to the Gröbner basis like (R=QQ[x,y,z]; fi=…; I=ideal(x*z-y^2,x^3-z^2); G=gb(I);f%G returning zero, $f_i\not\in I$. Division with respect to the elements in ideal […]

I would like to be able to draw any number of different shapes and determine the area of their intersections. I’m looking for free, open source software. I thought about trying to code something up myself, but it would save a lot of time and trouble if there is something out there that can do […]

I’m looking for free software to create equations, and then be able to export to formats such as .jpg, .pdf, or maybe even MathML. Open source is nice but not required, internet based would be great but a local installed application would work as well. Being able to create graphs would be a positive, but […]

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